What are the functional relationships?

Very detailed, see for yourself.

The concept and nature of 1 function (summary of knowledge points), a compulsory course of mathematics in the college entrance examination in 2006.

(A) the concept of function

The concept of 1. function: Let a and b be non-empty number sets. If any number X in set A has a unique number f(x) corresponding to it according to a certain correspondence F, then F: A → B is called a function from set A to set B, and is denoted as y=f(x). The value of y corresponding to the value of x is called the function value, and the set of function values {f(x)| x∈A} is called the range of the function.

Note: ○2 If only the analytical formula y=f(x) is given without specifying its domain, the domain of the function refers to the set of real numbers that can make the formula meaningful; ○3 The definition domain and value domain of a function should be written in the form of sets or intervals.

Domain supplement

The set of real numbers x that can make a function meaningful is called the domain of the function. The main basis for finding the domain of function is that the denominator of (1) score is not equal to zero; (2) The number of even roots is not less than zero; (3) The truth value of the logarithmic formula must be greater than zero; (4) Exponential radix and logarithmic radix must be greater than zero and not equal to 1. (5) If a function is composed of some basic functions through four operations, then its domain is a set of values of x that make all parts meaningful. (6) Exponential radix cannot be equal to zero. (6) The definition domain of function in practical problems should also ensure that practical problems are meaningful.

(Also note that finding the solution set of inequality group is the domain of function. )

2. The three elements of a function: definition domain, correspondence relationship and value domain.

Note again: (1) The three elements that make up a function are domain, correspondence and value. Because the range is determined by the domain and the corresponding relationship, two functions are called equal (or the same function) if and only if their domain and the corresponding relationship are exactly the same, but the independent variables and function values are represented by letters. The judgment method of the same function: ① the expressions are the same; (2) Domain consistency (two points must be met at the same time)

(See related example 2 on page 2 1 of the textbook)

Value range supplement

(1), the range of a function depends on the domain and the corresponding laws. No matter what method is adopted to find the range of a function, the domain must be considered first. (2) You should be familiar with the range of linear function, quadratic function, exponential function, logarithmic function and trigonometric function, which is the basis for solving the numerical range of reply. (3) The common methods to find the function range are: direct method and inverse function method.

3. Function image knowledge induction

(1) Definition: In the plane rectangular coordinate system, the set c of points P(x, y) with functions y = f (x) and (x ∈ a) as abscissa and function y as ordinate is called the image of functions y = f (x) and (x ∈ a).

The coordinates (x, y) of each point on c satisfy the functional relationship y=f(x). On the other hand, the points (x, y) whose coordinates are x and y for each group of ordered real numbers satisfying y=f(x) are all on c, that is, c = {p (x, y) | y = f (x).

Image c is generally a smooth and continuous curve (or straight line), or it may be composed of several curves or discrete points, and it has at most one intersection with any straight line parallel to the Y axis.

(2) Painting

A. Point tracing method: according to the resolution function and the definition domain, find some corresponding values of x and y and list them, trace the corresponding points p (x, y) in the coordinate system with (x, y) as coordinates, and finally connect these points with smooth curves.

B, image transformation method (please refer to the compulsory 4 trigonometric function)

There are three commonly used transformation methods, namely translation transformation, expansion transformation and symmetry transformation.

(3) Function:

1, intuitively see the nature of the function; 2. Analyze the thinking of solving problems by combining numbers and shapes. Improve the speed of solving problems.

Find mistakes in solving problems.

4. Understand the concept of interval.

Classification of (1) interval: open interval, closed interval and semi-open and semi-closed interval; (2) Infinite interval; (3) The number axis representation of the interval.

5. What is mapping?

Generally speaking, let A and B be two nonempty sets. If any element X in set A has a unique element Y corresponding to it according to a corresponding rule F, then the corresponding F: A B is the mapping from set A to set B ... Write it as "f: a b"

Given a mapping from set a to set b, if A ∈ A, B ∈ B and element a correspond to element b, then we call element b the image of element a and element a the original image of element B.

Note: Function is a special mapping, and mapping is a special correspondence. ① Set A, B and corresponding rule F are definite; (2) The correspondence rule is directional, that is, it emphasizes the correspondence from set A to set B, which is generally different from the correspondence from b to a; ③ For mapping F: A → B, it should be satisfied that: (i) every element in set A has an image in set B, and the image is unique; (ii) Different elements in set A and corresponding images in set B can be the same; (iii) Each element in set B does not need to have an original image in set A. ..

6. Common function representations and their respective advantages:

○ 1 function images can be continuous curves, straight lines, broken lines, discrete points, etc. Pay attention to the basis of judging whether a graph is a function image; ○2 Analysis method: the definition domain of the function must be indicated; ○3 mirror image method: attention should be paid to drawing by tracing point method: determine the definition domain of function; Simplify the analytical formula of the function; Observe the characteristics of the function; List method: the selected independent variables should be representative and reflect the characteristics of the field.

Note: Analytical method: it is convenient to calculate the function value. List method: it is easy to find the function value. Mirror image method: convenient to measure function value

Supplement 1: piecewise function (see textbook P24-25)

There are different functions in different parts of the domain to parse the expression. When finding the function value in different ranges, the independent variable must be substituted into the corresponding expression. The analytic expression of piecewise function cannot be written as several different equations. Instead, write several different expressions of function values and enclose them in left brackets, indicating the values of independent variables of each part respectively. (1) piecewise function is one function, so don't mistake it for several functions. (2) The definition domain of piecewise function is the union of the definition domain of each segment, and the value domain is the union of the value domain of each segment.

Supplement 2: Composite Function

If y=f(u), (u∈M), u=g(x), (x∈A), then y=f[g(x)]=F(x), (x∈A) is called the composite function of f and g.

For example: y=2sinX y=2cos(X2+ 1)

7. Monotonicity of functions

(1). Incremental function

Let the domain of function y=f(x) be I, if for any two independent variables x 1, x2 is in an interval d within the domain I, when X 1

If the values of any two independent variables in the interval d are both x 1, x2, when X 1 f (x2), then f (x) is said to be a decreasing function in this interval. The interval d is called monotonically decreasing interval y=f(x).

Note: the monotonicity of the function ○ 1 is a property on an interval within the definition domain and a local property of the function;

2 must be for any two independent variables in the interval d x 1, x2; When x 1

(2) the characteristics of image

If the function y=f(x) is increasing function or subtraction function in a certain interval, it is said that the function y=f(x) has (strict) monotonicity in this interval, and the image of increasing function rises from left to right, and the image of subtraction function falls from left to right.

(3) The method of judging monotone interval and monotonicity of function.

(1) Definition method:

○ 1 Let x 1, x2∈D, x 1

(b) Image method (rising and falling from image) _

The monotonicity of the compound function f[g(x)] is closely related to the monotonicity of its constituent functions u=g(x) and y=f(u), and its laws are as follows:

Monotonicity of function

U=g(x) increase or decrease.

Y=f(u) increase decrease increase decrease.

Y=f[g(x)] increase, decrease, decrease.

Note: 1, the monotone interval of the function can only be a subinterval of its domain, and the intervals with the same monotonicity cannot be summed together to write its union. 2. Do you remember the simple derivative method of judging monotonicity that we learned in elective courses?

8. Parity of functions

(1) even function

Generally speaking, f (-x) = f(x) exists for any x in the domain of function f(x), so f (x) is called even function.

(2) odd function

Generally speaking, F (-X) = f(x) exists for any X in the domain of function f(x), so F (X) is called odd function.

Note: ○ 1 If the function is odd function or even, it is called the parity of the function, and the parity of the function is the global property of the function; A function may have no parity, or it may be both a odd function and an even function.

○2 According to the definition of function parity, a necessary condition for a function to have parity is that -x must also be an independent variable in the domain for any x in the domain (that is, the domain is symmetric about the origin).

(3) Features of images with parity function

The image of even function is symmetrical about y axis; Odd function's image is symmetrical about the origin.

Summary: Using the format steps of defining and judging the parity of a function: ○ 1 First, determine the domain of the function and judge whether its domain is symmetrical about the origin; 2 determine the relationship between f (-x) and f(x); ○3. Draw the corresponding conclusion: if f (-x) = f(x) or f (-x)-f (x) = 0, then f(x) is an even function; If f (-x) =-f(x) or f (-x)+f (x) = 0, then f(x) is odd function.

Note: the symmetry of the function definition domain about the origin is a necessary condition for the function to have parity. First, whether the domain of the function is symmetric about the origin, and if not, whether the function is odd or even. If it is symmetric, (1) will be judged according to the definition. (2)f(-x)= f(x) is sometimes difficult to judge. We can consider whether f (-x) f (x) = 0 or F (x)/F (-x) =1; (3) Using the image judgment of theorems or functions.

9. Analytic expression of function

(1). The analytic formula of the function is a representation of the function. When the functional relationship between two variables is needed, the corresponding law between them and the definition domain of the function are needed.

(2) The main methods to find resolution function are: undetermined coefficient method, method of substitution method, parameter elimination method, etc. If the structure of the resolution function is known, the undetermined coefficient method can be used; When the expression of the compound function f[g(x)] is known, method of substitution can be used, so we should pay attention to the value range of elements; When the known expression is simple, the matching method can also be used; If the expression of abstract function is known, f(x) is usually obtained by solving equations and eliminating parameters.

10. Maximum (minimum) value of the function (see textbook p36 for definition).

○ 1 Finding the maximum (minimum) value of a function by using the properties of a quadratic function (matching method) ○ 02 Finding the maximum (minimum) value of a function by using an image ○ 03 Judging the maximum (minimum) value of a function by using the monotonicity of the function: If the function y=f(x) monotonically increases in the interval [a, b], in the interval [b,

1 1. There are two keys to solving mathematical application problems:

First, read the questions carefully, carefully examine the questions, accurately understand the meaning of the questions, make clear the actual background of the questions, and then scientifically abstract the actual problems into corresponding mathematical problems;

Second, it is necessary to reasonably select parameter variables. After the variables are set, it is necessary to find their internal relations, choose appropriate algebraic expressions to express the relationship in the problem, and establish corresponding mathematical models such as functions, equations and inequalities. Finally, the mathematical model is solved to solve practical problems.

Properties of function and characteristics of function image

Functional attributes define image features.

Function image

It is generally a continuous curve, or it may be composed of several curves or discrete points.

Defines the range of the independent variable x of the domain m, which exists on the left and right of the image.

Range n Function value y Range up and down the image.

uneven

idol

nature

odd function

Yes, any one.

F(-x)=-f(x) The image is symmetrical about the origin.